MPC of constrained discrete-time linear periodic systems — A framework for asynchronous control: Strong feasibility, stability and optimality via periodic invariance

State-feedback model predictive control (MPC) of discrete-time linear periodic systems with time-dependent state and input dimensions is considered. The states and inputs are subject to periodically time-dependent, hard, convex, polyhedral constraints. First, periodic controlled and positively invariant sets are characterized, and a method to determine the maximum periodic controlled and positively invariant sets is derived. The proposed periodic controlled invariant sets are then employed in the design of least-restrictive strongly feasible reference-tracking MPC problems. The proposed periodic positively invariant sets are employed in combination with well-known results on optimal unconstrained periodic linear-quadratic regulation (LQR) to yield constrained periodic LQR control laws that are stabilizing and optimal. One motivation for systems with time-dependent dimensions is efficient control law synthesis for discrete-time systems with asynchronous inputs, for which a novel modeling framework resulting in low dimensional models is proposed. The presented methods are applied to a multirate nano-positioning system.     

线性控制系统的无界多面体不变集

本文利用非光滑分析方法,讨论了线性控制系统的无界多面体不变集问题.当无界多面体的极方向满足一定条件时,得到了该无界多面体为一类线性控制系统弱不变集的判别方法.然后在更一般的线性控制系统下给出了无界多面体为强不变集的充分条件.最后给出两个应用实例.     

Constrained MPC for uncertain linear systems with ellipsoidal target sets

Robustly feasible invariant sets provide a way of identifying stabilizable regions for uncertain/time-varying linear systems with input constraints under fixed state feedback control laws. With the introduction of extra degrees of freedom in the form of perturbed control laws, these stabilizable regions can be enlarged. This was done in Lee and Kouvaritakis (Automatica 36 (2000) 1497–1504) in conjunction with polyhedral invariant sets and the aim here is to extend this work using ellipsoidal target sets. We also extend the analysis to take into account both polytopic and unstructured bounded disturbances, as well as unstructured uncertainties.     

Unconstrained and constrained stabilisation of bilinear discrete-time systems using polyhedral Lyapunov functions

The constrained and unconstrained stabilisation problem of discrete-time bilinear systems is investigated. Using polyhedral Lyapunov functions, conditions for a polyhedral set to be both positively invariant and domain of attraction for systems with second-order polynomial nonlinearities are first established. Then, systematic methods for the determination of stabilising linear feedback for both constrained and unconstrained bilinear systems are presented. Attention is drawn to the case where no linear control law rendering the pre-specified desired domain of attraction positively invariant exists. For this case, an approach guaranteeing the existence of a possibly suboptimal solution is established.     

Invariant cones for non-smooth dynamical systems

In this paper we study the dynamics of piecewise linear systems with regard to generalisations of the familiar notion of Hopf bifurcation for smooth systems. It is shown that the concept of invariant planes with a change from a stable focus to an unstable focus through a center should be replaced in the case of non-smooth systems by an invariant cone with similar dynamical behaviour. In that way results for planar systems are extended into higher dimensions. To determine the invariant cone and its stability properties various extended systems are derived.     

Positively invariant polyhedral sets of discrete-time linear systems

The problem of the existence of positively invariant polyhedral sets for linear discrete-time dynamical systems is studied. In the first part of the paper, necessary and sufficient conditions for a given polyhedral set to be a positively invariant set of a linear system are obtained. Then, the spectral properties of systems possessing this kind of invariant set are established. Finally the class of systems possessing positively invariant polyhedral cones is studied.     

A distributed model predictive control scheme for leader–follower multi-agent systems

In this paper, we present a novel receding horizon control scheme for solving the formation problem of leader–follower configurations. The algorithm is based on set-theoretic ideas and is tuned for agents described by linear time-invariant (LTI) systems subject to input and state constraints. The novelty of the proposed framework relies on the capability to jointly use sequences of one-step controllable sets and polyhedral piecewise state-space partitions in order to online apply the ‘better’ control action in a distributed receding horizon fashion. Moreover, we prove that the design of both robust positively invariant sets and one-step-ahead controllable regions is achieved in a distributed sense. Simulations and numerical comparisons with respect to centralised and local-based strategies are finally performed on a group of mobile robots to demonstrate the effectiveness of the proposed control strategy.     

Componentwise ultimate bound and invariant set computation for switched linear systems

We present a novel ultimate bound and invariant set computation method for continuous-time switched linear systems with disturbances and arbitrary switching. The proposed method relies on the existence of a transformation that takes all matrices of the switched linear system into a convenient form satisfying certain properties. The method provides ultimate bounds and invariant sets in the form of polyhedral and/or mixed ellipsoidal/polyhedral sets, is completely systematic once the aforementioned transformation is obtained, and provides a new sufficient condition for practical stability. We show that the transformation required by our method can easily be found in the well-known case where the subsystem matrices generate a solvable Lie algebra, and we provide an algorithm to seek such transformation in the general case. An example comparing the bounds obtained by the proposed method with those obtained from a common quadratic Lyapunov function computed via linear matrix inequalities shows a clear advantage of the proposed method in some cases.     

具有时变不确定线性系统的鲁棒无源控制

研究具有时变不确定参数的线性系统在有界能量外部输入作用下的鲁棒无源控制问题. 目的是设计一个反馈控制器使得闭环系统是二次稳定,同时具有严格无源性.研究结果证明,这 一问题可以转化为一个等价的具有某个参数的线性时不变系统的正实控制问题,从而可用现有的 正实控制问题的求解方法解决所研究的问题.     

An eigenstructure assignment approach for constrained linear continuous-time singular systems

This paper establishes necessary and sufficient algebraic conditions for positive invariance of convex polyhedra with respect to some linear continuous-time singular systems. They can be considered as an extension for linear singular systems of the classical positive invariance relations for regular linear systems. For a stabilizable and impulse controllable singular system with constrained inputs, a stabilizing state feedback control guaranteeing the closed-loop positive invariance of some polyhedral sets determined from the feedback matrix is studied. An analysis of the closed-loop positive invariance relations is thus presented in terms of eigenstructure and stability properties. An eigenstructure assignment technique is proposed depending on the number of stable finite poles of the singular system.     

Polyhedral Flows in Hybrid Automata

A hybrid automaton is a mathematical model for hybrid systems, which combines, in a single formalism, automaton transitions for capturing discrete updates with differential constraints for capturing continuous flows. Formal verification of hybrid automata relies on symbolic fixpoint computation procedures that manipulate sets of states. These procedures can be implemented using boolean combinations of linear constraints over system variables, equivalently, using polyhedra, for the subclass of linear hybrid automata. In a linear hybrid automaton, the flow at each control mode is given by a rate polytope that constrains the allowed values of the first derivatives. The key property of such a flow is that, given a state-set described by a polyhedron, the set of states that can be reached as time elapses, is also a polyhedron. We call such a flow a polyhedral flow. In this paper, we study if we can generalize the syntax of linear hybrid automata for describing flows without sacrificing the polyhedral property. In particular, we consider flows described by origin-dependent rate polytopes, in which the allowed rates depend, not only on the current control mode, but also on the specific state at which the mode was entered. We identify necessary and sufficient conditions for a class of flows described by origin-dependent rate polytopes to be polyhedral. We also propose and study additional classes of flows: strongly polyhedral flows, in which the set of states that can be reached up to a given time starting from a polyhedron is guaranteed to be a polyhedron, and polyhedrally sliced flows, in which the set of states that can be reached at a given time starting from a polyhedron is guaranteed to be a polyhedron. Finally, we discuss an application of the above classes of flows to approximate exponential behaviours.     

Piecewise quadratic Lyapunov functions over conical partitions for robust stability analysis

In this paper, the robust stability problem for uncertain linear continuous‐time systems is faced making use of piecewise quadratic Lyapunov functions (PQLF). PQLF are obtained by partitioning the state space into polyhedral cones and by associating a quadratic form with each cone. The proposed formulation allows us to recover, as particular cases of PQLFs, not only the class of quadratic functions but also the class of polyhedral functions. In this way, we manage to show the universality of the class of PQLF for the robust stability problem. The main contribution of the paper is the formulation of a low‐computational cost procedure for the stability analysis of uncertain linear systems. Several numerical examples are included in the paper, where the proposed approach is tested on some benchmark cases taken from the literature. Comparisons with existing methods show that the proposed method performs better under several aspects. Copyright © 2014 John Wiley & Sons, Ltd.     

基于最小衰减率多面体不变集的鲁棒模型预测控制

针对一类具有输入输出约束的多胞体结构线性变参数系统,提出了一种基于最小衰减率多面体不变集的鲁棒模型预测控制算法,算法分为在线和离线两个部分.为增强系统控制效果,提高系统响应速度,离线算法首先采用寻求状态变量的最小衰减率的方法优化出一系列状态变量及相应的状态反馈控制律,然后构建出相应的多面体不变集序列;在线算法根据当前实测状态变量,在多面体不变集序列内确定状态变量所处的最小多面体不变集,通过在线优化得出系统的控制输入.给出了鲁棒模型预测控制算法的详细步骤和系统的闭环稳定性证明.仿真结果验证了本算法的有效性,表明本算法使系统的闭环响应更为快速和稳定.     

A Lyapunov method for stability analysis of piecewise-affine systems over non-invariant domains

This paper analyses stability of discrete-time piecewise-affine systems, defined on possibly non-invariant domains, taking into account the possible presence of multiple dynamics in each of the polytopic regions of the system. An algorithm based on linear programming is proposed, in order to prove exponential stability of the origin and to find a positively invariant estimate of its region of attraction. The results are based on the definition of a piecewise-affine Lyapunov function, which is in general discontinuous on the boundaries of the regions. The proposed method is proven to lead to feasible solutions in a broader range of cases as compared to a previously proposed approach. Two numerical examples are shown, among which a case where the proposed method is applied to a closed-loop system, to which model predictive control was applied without a-priori guarantee of stability.     

Optimized robust control invariance for linear discrete-time systems: Theoretical foundations

This paper introduces the concept of optimized robust control invariance for discrete-time linear time-invariant systems subject to additive and bounded state disturbances. A novel characterization of two families of robust control invariant sets is given. The existence of a constraint admissible member of these families can be checked by solving a single and tractable convex programming problem in the generic linear-convex case and a standard linear/quadratic program when the constraints are polyhedral or polytopic. The solution of the same optimization problem yields the corresponding feedback control law that is, in general, set-valued. A procedure for selection of a point-valued, nonlinear control law is provided.     

A model predictive control framework for constrained uncertain positive systems

In this paper, a new model predictive control framework is proposed for positive systems subject to input/state constraints and interval/polytopic uncertainty. Instead of traditional quadratic performance index, simple linear performance index, linear Lyapunov function, cone invariant set with linear form and linear computation tool are first adopted. Then, a control law that can handle the constraints and robustly stabilise the systems is proposed. The advantages of the new framework lie in the following facts: (1) an equivalent linear problem is formulated that can be easily solved than other problems including the quadratic ones, (2) simple linear index and linear tool can be used based on the essential property of positive systems to achieve the desired control performance and (3) a general model predictive control law without sign restriction is designed. Finally, an attempt of application on mitigating viral escape is provided to verify the effectiveness of the proposed approach.     

Offline Robust Model Predictive Control for Lipschitz Non‐Linear Systems Using Polyhedral Invariant Sets

In this paper the concept of maximal admissible set (MAS) for linear systems with polytopic uncertainty is extended to non‐linear systems composed of a linear constant part followed by a non‐linear term. We characterize the maximal admissible set for the non‐linear system with unstructured uncertainty in the form of polyhedral invariant sets. A computationally efficient state‐feedback RMPC law is derived off‐line for Lipschitz non‐linear systems. The state‐feedback control law is calculated by solving a convex optimization problem within the framework of linear matrix inequalities (LMIs), which leads to guaranteeing closed‐loop robust stability. Most of the computational burdens are moved off‐line. A linear optimization problem is performed to characterize the maximal admissible set, and it is shown that an ellipsoidal invariant set is only an approximation of the true stabilizable region. This method not only remarkably extends the size of the admissible set of initial conditions but also greatly reduces the on‐line computational time. The usefulness and effectiveness of the method proposed here is verified via two simulation examples.     

A note on finite memory in locally square integrable systems

In the past decade J.C. Willems has developed the notion of the behavior of a system: the behavior of a system is the set of trajectories satisfying the system laws. He has studied especially linear time invariant discrete time systems in great detail. In this paper we concentrate on continuous time systems, in particular we study a class of locally square integrable systems and we investigate whether closed, linear, time invariant systems have finite memory.     

可变多胞体结构的鲁棒模型预测控制

针对一类具有输入输出约束的线性变参数系统,提出了一种可变多胞体结构的多面体不变集鲁棒模型预测控制算法,算法分为在线和离线两个部分。离线算法构建系统的未知但有界误差描述模型,根据状态的变化得到可变参数的多胞体结构,并转化为状态空间模型的多胞体形式,然后采用线性矩阵不等式的优化方式得到一系列离线嵌套多面体不变集。在线算法根据状态变量在嵌套多面体不变集中的位置,结合可变多胞体结构,通过线性插值的优化方式得到系统的实际控制律。给出算法的详细步骤和系统闭环稳定性证明。仿真结果验证了该算法的有效性,表明该算法使系统的闭环响应更为快速和稳定。     

Robust asymptotic stabilizability of Petersen's counterexample viaa linear static controller

A stabilizing linear static controller is discovered for a linear time-varying uncertain system of Petersen. We use the concept of a sandwich cone decomposition to show that Petersen's system is stabilizable by linear static controllers. We establish the surprising result that Petersen's system is stabilizable against time-varying uncertainties on any given compact subset of the uncertainty controllability space. Our work suggests the use of “scalar-quadratic” Lyapunov functions, a special subclass of polyhedral Lyapunov functions, in establishing stability for other linear time-varying uncertain systems, and it reopens the study of the conjecture: stabilizability via nonlinear control always implies stabilizability via linear control